L(s) = 1 | − 5-s − 7-s − 6·11-s − 13-s − 4·19-s + 25-s + 4·31-s + 35-s + 4·37-s + 6·41-s − 10·43-s + 6·47-s + 49-s − 6·53-s + 6·55-s + 10·61-s + 65-s − 4·67-s − 6·71-s + 2·73-s + 6·77-s + 4·79-s − 6·83-s + 6·89-s + 91-s + 4·95-s − 10·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s − 0.917·19-s + 1/5·25-s + 0.718·31-s + 0.169·35-s + 0.657·37-s + 0.937·41-s − 1.52·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.809·55-s + 1.28·61-s + 0.124·65-s − 0.488·67-s − 0.712·71-s + 0.234·73-s + 0.683·77-s + 0.450·79-s − 0.658·83-s + 0.635·89-s + 0.104·91-s + 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8870505958\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8870505958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.77101689271500, −12.50964357995367, −11.91717971801339, −11.34640520882760, −10.99287671341601, −10.46639727965054, −10.03293926869338, −9.794262459834751, −8.965374797911238, −8.592221728126880, −8.062908911408016, −7.732735226499886, −7.210936511022813, −6.750355736753226, −6.013036882020029, −5.807709554813855, −4.957983595864695, −4.731834197016309, −4.154455006023339, −3.426293446627989, −2.983358607123210, −2.392790026206363, −2.004521185420176, −0.9393362447629024, −0.2956808581706575,
0.2956808581706575, 0.9393362447629024, 2.004521185420176, 2.392790026206363, 2.983358607123210, 3.426293446627989, 4.154455006023339, 4.731834197016309, 4.957983595864695, 5.807709554813855, 6.013036882020029, 6.750355736753226, 7.210936511022813, 7.732735226499886, 8.062908911408016, 8.592221728126880, 8.965374797911238, 9.794262459834751, 10.03293926869338, 10.46639727965054, 10.99287671341601, 11.34640520882760, 11.91717971801339, 12.50964357995367, 12.77101689271500